Average Error: 3.6 → 2.4
Time: 47.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 7.876427020943232 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{\sqrt{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}}{\frac{\left(\beta + \alpha\right) + 2}{\frac{\sqrt{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}}{\left(\beta + \alpha\right) + 2}}}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 7.876427020943232 \cdot 10^{+161}:\\
\;\;\;\;\frac{\frac{\sqrt{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}}{\frac{\left(\beta + \alpha\right) + 2}{\frac{\sqrt{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}}{\left(\beta + \alpha\right) + 2}}}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\

\end{array}
double f(double alpha, double beta) {
        double r2889970 = alpha;
        double r2889971 = beta;
        double r2889972 = r2889970 + r2889971;
        double r2889973 = r2889971 * r2889970;
        double r2889974 = r2889972 + r2889973;
        double r2889975 = 1.0;
        double r2889976 = r2889974 + r2889975;
        double r2889977 = 2.0;
        double r2889978 = 1.0;
        double r2889979 = r2889977 * r2889978;
        double r2889980 = r2889972 + r2889979;
        double r2889981 = r2889976 / r2889980;
        double r2889982 = r2889981 / r2889980;
        double r2889983 = r2889980 + r2889975;
        double r2889984 = r2889982 / r2889983;
        return r2889984;
}
↓
double f(double alpha, double beta) {
        double r2889985 = alpha;
        double r2889986 = 7.876427020943232e+161;
        bool r2889987 = r2889985 <= r2889986;
        double r2889988 = 1.0;
        double r2889989 = beta;
        double r2889990 = r2889989 * r2889985;
        double r2889991 = r2889989 + r2889985;
        double r2889992 = r2889990 + r2889991;
        double r2889993 = r2889988 + r2889992;
        double r2889994 = sqrt(r2889993);
        double r2889995 = 2.0;
        double r2889996 = r2889991 + r2889995;
        double r2889997 = r2889994 / r2889996;
        double r2889998 = r2889996 / r2889997;
        double r2889999 = r2889994 / r2889998;
        double r2890000 = r2889996 + r2889988;
        double r2890001 = r2889999 / r2890000;
        double r2890002 = 1.0;
        double r2890003 = r2889988 / r2889985;
        double r2890004 = r2890002 - r2890003;
        double r2890005 = 2.0;
        double r2890006 = r2889985 * r2889985;
        double r2890007 = r2890005 / r2890006;
        double r2890008 = r2890004 + r2890007;
        double r2890009 = r2890008 / r2889996;
        double r2890010 = r2890009 / r2890000;
        double r2890011 = r2889987 ? r2890001 : r2890010;
        return r2890011;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 7.876427020943232e+161

    1. Initial program 1.3

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity1.3

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    4. Applied add-sqr-sqrt1.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    5. Applied times-frac1.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{1} \cdot \frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    6. Applied associate-/l*1.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{1}}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    if 7.876427020943232e+161 < alpha

    1. Initial program 16.2

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Taylor expanded around inf 8.0

      \[\leadsto \frac{\frac{\color{blue}{\left(2.0 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 1.0 \cdot \frac{1}{\alpha}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    3. Simplified8.0

      \[\leadsto \frac{\frac{\color{blue}{\frac{2.0}{\alpha \cdot \alpha} + \left(1 - \frac{1.0}{\alpha}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7.876427020943232 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{\sqrt{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}}{\frac{\left(\beta + \alpha\right) + 2}{\frac{\sqrt{1.0 + \left(\beta \cdot \alpha + \left(\beta + \alpha\right)\right)}}{\left(\beta + \alpha\right) + 2}}}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - \frac{1.0}{\alpha}\right) + \frac{2.0}{\alpha \cdot \alpha}}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) + 1.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))